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Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

Differential geometry is a branch of mathematics that uses techniques in calculus and linear algebra to study geometry. It is closely related to differential topology and PDEs (geometric analysis).

32835 questions
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Prove that curve with zero torsion is planar

I have proved that a planar curve of zero curvature is a straight line. It follows from the Frenet equations. But now I need to prove that if $\varkappa=0$, then the space curve $\mathbf{r}(t)$ is planar. From the condition and the Frenet equations…
Sergey Filkin
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Easy example of unit speed plane curve?

I was trying to find a non-trivial example of a unit speed plane curve. The reason is I want something to work with but if I start with a non-unit speed curve and then do the arc length parameterisation I end up with something impossible. The…
student
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Understanding composition of vector fields

I've finished a first course on differential geometry and I still find it confusing on how to compose/multiply two vector fields. Let's assume that $X$ and $Y$ are two vector fields on a smooth manifold. Then my understanding of $XY$ is that: $$XY f…
student
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Gradient vector derived from the metric tensor

According to Frankel's book "The Geometry of Physics", the components of a contravariant gradient vector can be obtained from the inverse of the metric tensor as follows (in section 2.1d, Page 73): $$ (\nabla f)^i = \sum_j g^{ij} \frac{\partial…
TJH
  • 129
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Example of orthogonal parametrization of a surface

I recently came to know about the orthogonal parametrization of a surface, for which $F={\bf X_u}\cdot{\bf X_v}=0$ and $E={\bf X_u}\cdot{\bf X_u}=G={\bf X_v}\cdot{\bf X_v}$. Here, $(E,F,G)$ denote the coefficients of the first fundamental form of a…
kolahalb
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Proving Gaussian curvature $\leq 0$ on line

Given some regular surface $S$ that contains a line $L$, I need to prove that the Gaussian curvature $K\leq 0$ at all points of $L$. I am thinking that if I could show that no points on $L$ can be elliptical $K>0$ then I would be set. But how can I…
user34166
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What's wrong with my osculating paraboloid?

I'm trying to learn about the Second Fundamental Form and I thought it would be fun to set up a surface in Geogebra and try to calculate the osculating paraboloid as I moved a point around on it, for certain values of "fun". I'm using Banchoff &…
helveticat
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Why is $\deg(f)$ well-defined?

If $M$ and $N$ are boundaryless, compact, connected, oriented $n$-manifolds, and $f:M\to N$ is smooth, then if $\omega_0$ is a $n$-form on $N$ and $\int_N\omega_0\neq 0$, there is a number $a$ such that $\int_M f^*\omega_0=a\int_N\omega_0$. In fact,…
Jrrow
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Computing the exterior derivative of a wedge product

How can we prove the following relation for differentiating the wedge product of a p-form $\alpha_p$ and a q-form $\beta_q$? $$d(\alpha_p\wedge\beta _q)=d\alpha_p\wedge\beta_q+(-1)^{p}\alpha_p\wedge d\beta_q$$
Roshan Shrestha
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Curve from curvature

It is possible to obtain the parameters of a curve in 2d simply by having only its curvature k(s)? I need to obtain its parametric equations in order to reconstruct the curve but i don't have any idea how or even if its possible? I try to search on…
metamail
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Why are derivations useful for defining tangent vectors?

On page 54 in his book Introduction to Smooth Manifolds, John Lee says the following: A linear map $v: C^\infty (M) \rightarrow \Bbb{R}$ is called a derivation at p if it satisfies \begin{equation} v(fg) = f(p)vg + g(p)vf \end{equation} for all…
Stan Shunpike
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Differential form is closed if the integral over a curve is rational number.

The following problem comes from do Carmo's book Differential Forms and Applications, Chapter 2, Exercise 4: Let $\omega$ be a differentiable 1-from defined on an open subset $U\subset \mathbb{R}^n$. Assume that for each closed differential curve…
Junyu
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An inequality about curvature

Let $\alpha:(a,b)\rightarrow\mathbb R^2$ be a regular parametrized plane curve. Assume that there exists $t_0$, $a
Vladimir
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What is some good intuition for being parallel or Lie derivative is zero?

I would like to know, if my unterstanding of the difference between parallelness and zero Lie derivative so far are accurate: \ 1. Consider a Riemannian manifold $M$. My understanding of $L_vT=0$ for some tensor field $T$ and $L_v$ being the Lie…
nick
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Equivariant differential forms.

I have some question about the equivariant differential forms on a smooth manifold: \ The equivariant differential forms over some smooth manifold $M$, on which the compact Lie group $G$ acts, are defined to be $$ \Omega_{G}^q(M)= \oplus_{2i+j=q}…
nick
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